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Question
If n +2C8 : n − 2P4 = 57 : 16, find n.
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Solution
\[\Rightarrow \frac{{}^{n + 2} C_8}{{}^{n - 2} P_4} = \frac{57}{16}\]
\[ \Rightarrow \frac{(n + 2)!}{8! (n - 6)!} \times \frac{(n - 6)!}{(n - 2)!} = \frac{57}{16}\]
\[ \Rightarrow \frac{(n + 2) (n + 1) n (n - 1) (n - 2)!}{8!} \times \frac{1}{(n - 2)!} = \frac{57}{16}\]
\[ \Rightarrow (n + 2) (n + 1) n (n - 1) = \frac{57}{16} \times 8! = \frac{19 \times 3}{16} \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\]
\[ \Rightarrow (n + 2) (n + 1) n (n - 1) = 143640\]
\[ \Rightarrow (n - 1) n (n + 1) (n + 2) = 19 \times 3 \times 7 \times 6 \times 5 \times 4 \times 3\]
\[ \Rightarrow (n - 1) n (n + 1) (n + 2) = 19 \times (3 \times 7) \times (6 \times 3) \times (4 \times 5)\]
\[ \Rightarrow (n - 1) n (n + 1) (n + 2) = 18 \times 19 \times 20 \times 21\]
\[ \Rightarrow n - 1 = 18\]
\[ \Rightarrow n = 19\]
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